Average Error: 31.2 → 0.3
Time: 25.5s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.47606870075120832908055490406695753336:\\ \;\;\;\;\left(\log \left(e^{\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}}\right) + \left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + 1\right)\right) - \frac{\sin x}{\cos x} \cdot \frac{\sin x}{x \cdot x}\\ \mathbf{elif}\;x \le 4.737990132830219636161928065121173858643:\\ \;\;\;\;\left(\left(x \cdot \frac{9}{40}\right) \cdot x - \frac{27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(e^{\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}}\right) + \left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + 1\right)\right) - \frac{\sin x}{\cos x} \cdot \frac{\sin x}{x \cdot x}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.47606870075120832908055490406695753336:\\
\;\;\;\;\left(\log \left(e^{\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}}\right) + \left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + 1\right)\right) - \frac{\sin x}{\cos x} \cdot \frac{\sin x}{x \cdot x}\\

\mathbf{elif}\;x \le 4.737990132830219636161928065121173858643:\\
\;\;\;\;\left(\left(x \cdot \frac{9}{40}\right) \cdot x - \frac{27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) - \frac{1}{2}\\

\mathbf{else}:\\
\;\;\;\;\left(\log \left(e^{\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}}\right) + \left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + 1\right)\right) - \frac{\sin x}{\cos x} \cdot \frac{\sin x}{x \cdot x}\\

\end{array}
double f(double x) {
        double r714143 = x;
        double r714144 = sin(r714143);
        double r714145 = r714143 - r714144;
        double r714146 = tan(r714143);
        double r714147 = r714143 - r714146;
        double r714148 = r714145 / r714147;
        return r714148;
}
↓
double f(double x) {
        double r714149 = x;
        double r714150 = -2.4760687007512083;
        bool r714151 = r714149 <= r714150;
        double r714152 = sin(r714149);
        double r714153 = r714152 / r714149;
        double r714154 = cos(r714149);
        double r714155 = r714153 / r714154;
        double r714156 = r714155 - r714153;
        double r714157 = exp(r714156);
        double r714158 = log(r714157);
        double r714159 = r714152 / r714154;
        double r714160 = r714159 / r714149;
        double r714161 = r714160 * r714160;
        double r714162 = 1.0;
        double r714163 = r714161 + r714162;
        double r714164 = r714158 + r714163;
        double r714165 = r714149 * r714149;
        double r714166 = r714152 / r714165;
        double r714167 = r714159 * r714166;
        double r714168 = r714164 - r714167;
        double r714169 = 4.73799013283022;
        bool r714170 = r714149 <= r714169;
        double r714171 = 0.225;
        double r714172 = r714149 * r714171;
        double r714173 = r714172 * r714149;
        double r714174 = 0.009642857142857142;
        double r714175 = r714165 * r714165;
        double r714176 = r714174 * r714175;
        double r714177 = r714173 - r714176;
        double r714178 = 0.5;
        double r714179 = r714177 - r714178;
        double r714180 = r714170 ? r714179 : r714168;
        double r714181 = r714151 ? r714168 : r714180;
        return r714181;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -2.4760687007512083 or 4.73799013283022 < x

    1. Initial program 0.0

      \[\frac{x - \sin x}{x - \tan x}\]

    2. Taylor expanded around inf 0.4

      \[\leadsto \color{blue}{\left(1 + \left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {x}^{2}} + \frac{\sin x}{\cos x \cdot x}\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{\cos x \cdot {x}^{2}}\right)}\]

    3. Simplified0.4

      \[\leadsto \color{blue}{\left(\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + 1\right) + \left(\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}\right)\right) - \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{\cos x}}\]
    4. Using strategy rm

    5. Applied add-log-exp0.4

      \[\leadsto \left(\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + 1\right) + \left(\frac{\frac{\sin x}{x}}{\cos x} - \color{blue}{\log \left(e^{\frac{\sin x}{x}}\right)}\right)\right) - \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{\cos x}\]

    6. Applied add-log-exp0.4

      \[\leadsto \left(\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + 1\right) + \left(\color{blue}{\log \left(e^{\frac{\frac{\sin x}{x}}{\cos x}}\right)} - \log \left(e^{\frac{\sin x}{x}}\right)\right)\right) - \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{\cos x}\]

    7. Applied diff-log0.4

      \[\leadsto \left(\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + 1\right) + \color{blue}{\log \left(\frac{e^{\frac{\frac{\sin x}{x}}{\cos x}}}{e^{\frac{\sin x}{x}}}\right)}\right) - \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{\cos x}\]

    8. Simplified0.4

      \[\leadsto \left(\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + 1\right) + \log \color{blue}{\left(e^{\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}}\right)}\right) - \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{\cos x}\]

    if -2.4760687007512083 < x < 4.73799013283022

    1. Initial program 62.9

      \[\frac{x - \sin x}{x - \tan x}\]

    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\]

    3. Simplified0.2

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \frac{9}{40}\right) - \frac{27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) - \frac{1}{2}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.47606870075120832908055490406695753336:\\ \;\;\;\;\left(\log \left(e^{\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}}\right) + \left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + 1\right)\right) - \frac{\sin x}{\cos x} \cdot \frac{\sin x}{x \cdot x}\\ \mathbf{elif}\;x \le 4.737990132830219636161928065121173858643:\\ \;\;\;\;\left(\left(x \cdot \frac{9}{40}\right) \cdot x - \frac{27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(e^{\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}}\right) + \left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + 1\right)\right) - \frac{\sin x}{\cos x} \cdot \frac{\sin x}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))